One way to predict the flow performance of subsurface oil and gas reservoirs is to solve differential equations corresponding to the physical laws that govern the movement of fluids in the subsurface. Because of the nature of the problem, the differential equations are conventionally solved using numerical methods working in discrete representations in space and time. Solving such equations typically requires the use of three dimensional, discrete representations of the subsurface rock properties and the associated fluids in the rocks.
In the oil and gas industry, numerical methods to solve for the flow of fluids in the reservoir are called “Numerical Reservoir Simulation”, or simply “Flow Simulation”. Predictions of future performance of subsurface oil and gas reservoirs with models based on physical laws are considered the highest standard in current technology. The three dimensional, discrete models of the subsurface are constructed in such a way that the models are consistent with actual measurements taken from the reservoir. Some of these measurements can be included directly in the model at the time of the construction. Other measurements, such as ones that are related to the movement of fluids within the reservoir, are used in an indirect manner utilizing a model calibration process. The calibration process involves assigning properties to the model and then verifying that the solutions computed with a numerical reservoir simulator are consistent with the measurements of the fluids. This calibration process is iterative and stops when the reservoir model is able to replicate the observations within a predetermined tolerance. Once the model is appropriately calibrated, the model can be run in a flow simulator to forecast or predict future performance.
The process of calibrating numerical models of oil and gas reservoirs to measurements related to production and/or injection of fluids is usually referred to as history matching. The calibration problem described previously may be considered as being a particular case within the field of inverse problem theory in mathematics. While there exists a rigorous mathematical framework for the solution of model calibration problems, such a framework becomes impractical for dealing with complex problems such as large scale reservoir flow simulation. For a detailed explanation of such a framework, see A. Tarantola, Inverse Problem Theory—Methods for Data Fitting and Model Parameter Estimation, Elsevier, 1987, hereinafter referred to as “Tarantola”. This Tarantola reference is hereby incorporated by reference in its entirety into this specification.
There are numerous difficulties in calibrating numerical models of oil and gas reservoirs to data related to the movement of fluids within the reservoirs. First, numerical models based on laws of physics are usually complex and a significant amount of computational time is required to evaluate, i.e. run a simulation on, each numerical model. Data to calibrate the numerical models are often uncertain. Furthermore, data to calibrate numerical models are scarce, both in time and space dimensions. Finally, there is not a unique solution to the calibration problem. Rather, there are many ways to calibrate a numerical model that is still consistent with all the measurements. Thus, there is not a unique calibrated numerical model. Accordingly, a probability is associated with any combination of model parameters and this probability may be expressed such as by using a probability density function (PDF).
The mathematical inverse problem theory provides the framework to deal with the inverse problem presented by reservoir flow simulation. Tarantola describes the mathematical theory applicable to the problem of calibration and uncertainty estimation. The solution to the problem is based on application of techniques relying on Monte Carlo simulation. The general approach prescribed by the mathematical theory, as described by Tarantola, can be summarized with a high level of simplification as follows.
A parameterization system, comprising model parameters, is defined for a mathematical model. Initially, an “a priori” probabilistic description is defined for the model parameters describing the mathematical model. Next, a probabilistic model is defined for measured or observed data which is to be used for calibration. This probabilistic model is constructed by defining a measure of the discrepancy between actual observed measurements of parameters and corresponding calculated parameters predicted by using the mathematical model. This measure of discrepancy is associated with a “likelihood” function in a Bayesian approach to updating probabilities. Then an “a posteriori” probabilistic description of the model parameters is constructed by updating the “a priori” probabilistic model using the observed measurements. In the most general case, the model parameter space is sampled in such a way that the resulting probability density function provides the desired “a posteriori” probabilistic description of the model parameters. The sampling takes into account the “a priori” model description. A common approach for performing the sampling is the application of variants of the Metropolis algorithm for Monte Carlo sampling. This process also produces probability density functions that correspond to the predictions calculated with the reservoir model.
The step of sampling the model parameter space is the most computational demanding part of this process and limits the practical application of this rigorous mathematical approach to solving problems involving oil and gas reservoir models based on physical laws. Using terminology commonly associated with inverse problem theory, the process involves solving the “forward problem” (running the flow simulation) a very large number of times during the sampling of the parameter space. The “forward problem” refers to computing the model response to a given combination of input model parameters.
Tarantola describes the use of probability theory in inverse problems such as in history matching and production forecasting. Likelihood functions need to be computed in the applications described by Tarantola. A likelihood function is a measure of how good results from a simulation run on a proposed model are as compared to actual observed values. Computation of likelihood functions in conjunction with very large models, such as are used in reservoir simulations, are not practical due to great computational costs. Evaluation of a likelihood function requires a reservoir simulation run. Each run of a large reservoir simulation may require hours of time to complete. Furthermore, thousands of such simulations may be required to obtain valid results.
There is a need for a practical method for history matching and forecasting wherein the high computational costs associated with calculating likelihood functions are reduced to a manageable level. The present invention addresses this need.